The elementary properties of generators of a ruled surface, and the existence of a line of striction when the surface is skew, are well known to readers of this journal. We propose to show that many of these properties do not belong exclusively to ruled surfaces; but that a family of curves on any surface possesses a line of striction and a focal curve or envelope, though these are not always real. When the surface is developable, and the “curves” are the generators of one system, the focal curve is the edge of regression. We shall also see that the properties to be established for a family of curves on a surface are analogous to some of the leading properties of congruences of curves in space, the line of striction corresponding to the surface of striction or orthocentric surface, and the focal curve to the focal surface of the latter.

Readers of the article in the October Magazine on “Systems of Conics satisfying four conditions” would notice the important part played by degenerate conics. A little more explanation of these degenerate conics may not be out of place.

The method of introducing e so common in text-books based upon an algebraic proof of the exponential series may be satisfactory for mathematical students, but it does not meet the needs of technical students (especially evening students), for it is impossible, if even desirable, to treat the matter along these lines in a manner at once brief, exact, and comprehensible. Moreover, it suffers from the great disadvantage that the property of e, which is of prime importance in technical applications, namely that ex is the function appropriate to the expression of the law of growth or decay so common in these—in other words, that ex is the solution of the differential equation dy/dx=y —appears as an afterthought, and not as the fundamentally important property, the raison d’être of the inclusion of e in the course and in the engineering formulae. The method outlined below, which has been tried for some few sessions, aims at presenting the differential equation as arising in the attempt to solve a concrete practical problem, and the discovery and identification of two solutions—one a series, the other in terms of indices. By obtaining the series first some idea of the use of non-terminating series, and appreciation of convergence, are gained. To simplify the numerical calculations, the index solution is then found. Here e appears as a limit, and to obtain its numerical value recourse must be had to the series.

The following lesson in Theoretical Arithmetic was originally intended as an Appendix to a Higher Arithmetic, and is now published in the hope that it may be of interest to the more select band of young scholars who may wish to know more of the subject than its application to the calculations of Stocks and Shares, et hoc omne genus.